Pressure-correlation phase diagram unraveling quantum phase transitions in altermagnetic $ε$-FeOOH and realization of piezo-anomalous Hall effect

We establish a pressure-correlation phase diagram that yields multiple pressure-tunable quantum phase transitions in iron oxyhydroxide ($ε$-FeOOH) across three critical pressures, Pc1>Pc2>Pc3. The transitions arise from the competition among electron-correlation, crystal-field, and Hund's coupling. First-principles calculations reveal that the earlier reported high-spin (HS) to low-spin (LS) crossover at correlation-sensitive Pc1 is accompanied by a transition from charge-transfer to Mott insulating (CTI-MI) phase, and a change in the spin-quantization axis (SQA) from ŷ to x̂. The HS-LS crossover and CTI-MI transition is driven by the simultaneous increase in crystal-field splitting and decrease in Hund's coupling. The critical pressure Pc2 governing the structural transition from centrosymmetric $Pnnm$ phase to noncentrosymmetric P2₁nm phase is found to be independent of electron-correlation. When correlation is strong, a third critical pressure (Pc3) emerges in the lower-pressure regime of the phase diagram. Validating the recent experimental observation, Pc3 is responsible for changing the SQA from ẑ to ŷ. By exhibiting altermagnetism in both P2₁nm and $Pnnm$ phases and possessing finite anomalous Hall conductivity (AHC) only for SQA along x̂ and ŷ, $ε$-FeOOH enables a unique piezo-anomalous Hall effect where one can switch on the AHC at a lower pressure and change its direction at a higher pressure.